This Sect. 7.3 shows su¢cient conditions for a relation to possess the weak W-property and the connection with Bosi and Zuanon (2017)’s quasi upper semicontinuity is clari…ed.
7.3.1 Quasi upper semicontinuous preorders
Recently, Bosi and Zuanon (2017) have introduced a notion of continuity that allows for a uni…cation of some results on the existence of a maximal element;
applications can be found also in Bosi and Zuanon (2019, Sect. 3). De…nition 8 recalls the notion of quasi upper semicontinuity. Proposition 10 shows that every quasi upper semicontinuous relation has the weak W-property.
De…nition 8 Let (X; ) be a topological space. A relation B on X is quasi upper semicontinuous i¤ B admits a quasi-extension R that is a preorder and that is closed-valued for .
Proposition 10 Let(X; )be a topological space andB be a relation on X. If B is quasi upper semicontinuous, thenB has the weak W-property for . Proof. Suppose B is quasi upper semicontinuous. Then B admits a quasi-extensionRthat is a preorder and that is closed-valued for . Part 1 of Propo-sition 3 ensures thatR is S-consistent and part 2 of Proposition 6 ensures that RisTt-closed-valued for . So,Rhas the W-property for . As every topology is compactly …ner than itself,B has the weak W-property for .
Remark 10 Let (X; )be a topological space and B be a preorder on X. The closed-valuedness of B for and the open-valuedness ofBca for are su¢cient conditions forB to be quasi upper semicontinuous: see Bosi and Zuanon (2017, Remark 2.5). Other su¢cient conditions are shown therein. In particular—and related to a result in Nosratabadi (2014)—Theorem 2.11 in Bosi and Zuanon (2017) asserts that, when is a second countable topology, the quasi upper semicontinuity of B is equivalent to the representability of Ba by means of an upper semicontinuous weak utility function (i.e., to the existence of an upper semicontinuous functionu:X !Rsuch that(y; x)2Ba )u(y)> u(x)).11
7.3.2 Lexicographic orders
Proposition 11 shows that the lexicographic order onR is an instance of a re-lation with the weak W-property for the natural topology ofR . Proposition 12 clari…es that, whenR is endowed with the natural topology, the lexicographic order onR is not, in general, a quasi upper semicontinuous preorder. Indeed, Proposition 12 proves the usefulness of the generalization of notion of a quasi upper semicontinuous preorder brought about by the weak W-property.
Proposition 11 Let be a non-zero ordinal—possibly, a positive integer—and endowR with the natural topology. The lexicographic order onR has the weak W-property for the natural topology of R .
Proof. Let 1denote the natural topology ofR and 2the lexicographic lower topology onR . Part 1 of Theorem 11 ensures that is closed-valued for 2: as is a preorder, from part 1 of Proposition 3 and part 2 of Proposition 6 we infer that has the W-property for 2. Part 2 of Theorem 11 ensures that
2 is compactly …ner than 1: as every relation is a quasi-extension of itself, has the weak W-property for 1.
Proposition 12 EndowR2 with the natural topology. The lexicographic order on R2 is a preorder but not a quasi upper semicontinuous preorder.
1 1For instance, endowingX= [0;2]with the subspace topology inherited from the natural topology of R, the preorder relation B on X de…ned by B(x) = [x;2] if x 1 and by B(x) = [x;1)ifx <1is quasi upper semicontinuous (in thatidX is a continuous weak utility forBa). It is worth to point out that, whenBis understood as a constant “variable preference relation” in the sense of Luc and Soubeyran (2013),Bis not “upper closed” in the sense of Luc and Soubeyran (2013, De…nition 10). In fact, Luc and Soubeyran (2013)’s upper closedness does not subsume Bosi and Zuanon (2017)’s quasi upper semicontinuity.
Proof. Let denote the natural topology of R2. It is known—and readily veri…ed—that 2 is a preorder and that 2 can be equivalently de…ned as the relation induced by the lexicographic cone
C= (R++ R)[(f0g R+).
Note thatC =Cnf(0;0)gand that a2(x) =x+C for allx2R2. By way of contradiction, assume the existence of a preorder relationRthat quasi-extends
2and that is closed-valued for . AsRquasi-extends 2, a2(x) Ra(x)for all x2R2; asRa R, we have thatRa(x) R(x)for allx2R2. So,x+C R(x) for all x2 R2 and basic properties of the topological closure operation entail that cl (x+C ) cl R(x)for allx2R2. Since the topology is translation invariant and sinceRis closed-valued for , we infer thatx+ cl C R(x)for allx2R2. Noting thatcl C =R+ R, we conclude that
x+ (R+ R) R(x)for allx2R2. (12) Put p= (0;0)and q= (0;1) and note that q2 a2(p). From (12) we conclude that q 2 R(p)and p2 R(q): a contradiction with the membership q 2 a2(p) and the assumption thatR quasi-extends 2.
7.3.3 Transitivity and closed-valuedness
A transitive and closed-valued relation need not have the W-property: this claim is readily veri…ed by considering the relation illustrated in Remark 4. Propo-sition 13 proves that any transitive and closed-valued relation has, however, the weak W-property (even when the topology is not T1). Recalling that the graph-closedness of a relation implies its closed-valuedness, it is thus clear that Theorem 12 subsumes Theorem I in Wallace (1962). Also, in the light of Propo-sition 2, it is thus clear that Theorem 12 subsumes PropoPropo-sition A.1 in Banks et al. (2006) (equivalently, Lemma 1 in Duggan (2011)).
Proposition 13 Let(X; )be a topological space andB be a relation on X. If B is transitive and closed-valued for , thenB has the weak W-property for . Proof. Put R = Br and 1 = . Let 0 denote the co…nite topology on X and let 2 be the topology generated by 0[ 1. Suppose B is transitive and closed-valued for 1: we show thatB has the weak W-property for 1. AsR is the re‡exive closure of the transitive relationB, the relationRis a preorder that quasi-extendsB. By Theorem 10, 2 is …ner than 1and compactly equivalent to 1; a fortiori, 2is compactly …ner than 1. AsB is closed-valued for 1 and
2 is …ner than 1, the relationB is closed-valued also for 2. By Theorem 10, the topology 2 is T1. As 2is a T1 topology andB is closed-valued for 2, its re‡exive closureR is closed-valued also for 2. Said this, part 1 of Proposition 3 and part 2 of Proposition 6 ensure thatRhas the W-property for 2. We are now in a position to conclude thatB has the weak W-property for 1.
An obvious—yet interesting—consequence of Proposition 13 and Theorem 12 is that the assumption of re‡exivity ofB in statements of Theorems 4 and 5 can be simply dropped. Corollary 6 provides a restatement of Theorems 5 dispensing with the unnecessary re‡exivity assumption imposed therein.
Corollary 6 Let (X; )be a topological space and B be a relation onX. If B is closed-valued for and transitive, thenM(B; Y)6=; for everyY 2 K(X; ).
7.3.4 Cones
Parts 1 and 2 of Proposition 14 show that the strong W-property is possessed by any relation induced by a convex cone of a real topological vector space that is either closed or open: the weak and the strong Pareto dominance relations on Rn are instances of economic interest.12 Part 3 of Proposition 14 shows that the—possibly neither transitive nor acyclic—relation induced by a strictly supported cone of a real topological vector space has the weak W-property. The fact that the relation induced by a strictly supported cone possesses a maximal element on every nonempty compact constraint set is a classic result of vector optimization: see, e.g., Luc (1989, Corollary 3.6).
Proposition 14 Let (X; )be a real topological vector space.
1. The relation induced by a -closed convex cone of X has the strong W-property for .
2. The relation induced by a -open convex cone of X has the strong W-property for .
3. The relation induced by a strictly -supported cone of X has the weak W-property for .
Proof. 1. SupposeC is a -closed convex cone ofX and letB be the relation induced by C. IfC is empty, thenB is the empty relation onX and the proof is obvious. Suppose C is nonempty. In a real topological vector space, any nonempty -closed cone contains the zero vector; also, the relation induced on a real vector space by a convex cone that contains the zero vector is a preorder. So, B is a preorder. As the topology of a real topological vector space is invariant under translation, the preorder B is closed-valued for . Said this, part 1 of Proposition 5 ensures thatRhas the strong W-property for .
2. Suppose C is a nonempty -open convex cone of X and let B be the relation induced by C. Recalling that the relation induced on a real vector space by a convex cone is transitive, we infer that B is transitive. As the topology of a real topological vector space is invariant under translation and scalar multiplication by a non-zero scalar, Bc is open-valued. Said this, part 2 of Proposition 5 ensures thatRhas the strong W-property for .
3. SupposeCis a cone ofX andH is a -open half-space includingC . Let B be the relation induced byC andRbe the relation induced byH. AsH is a -open convex cone ofX,Rhas the W-property for by part 2 of Proposition 14 and Proposition 7. Note that Ba and Ra are, respectively, the relations induced byC andH . AsC H =H , we infer thatRquasi-extendsB. As every topology is compactly …ner than itself, we are in a position to conclude that B has the weak W-property for .
Corollary 7 shows two consequences of Proposition 14. Remark 11 points out that a certain class of Bewley (justi…able) preferences lies within the class of relations with the strong (weak) W-property.
1 2Agreeing thatRnis the set of all conceivable utility levels of an economy withnagents, the weak (strong) Pareto dominance relation is the relation onRninduced by the nonnegative (positive) orthant ofRn, which is a closed (open) cone ofRn.
Corollary 7 Let(X; )be a real topological vector space.
1. The relation induced by the intersection of a family of -closed half-spaces ofX has the strong W-property for .
2. The relation induced by the union of a family of -closed half-spaces ofX has the weak W-property for .
Proof. 1. LetfHigi2I be a family of -closed half-spaces ofX. Put C=T
i2IHi
and letB be the relation induced byC. Note thatC is a -closed convex cone ofX. AsCis a -closed convex cone ofX,B has the strong W-property for by part 1 of Proposition 14.
2. LetfHigi2I be a family of -closed half-spaces ofX. Put C=S
i2IHi
and let B be the relation induced by C. IfI is the empty set, then B is the empty relation and the proof is obvious. Suppose I is nonempty and pick an arbitrary H 2 fHigi2I. Let R be the relation induced byH. From part 1 of Proposition 14 and Proposition 7 we infer thatRhas the W-property for . A moment’s re‡ection shows thatC H . Consequently,Ba Raand henceR is a quasi-extension ofB. Therefore,B has the weak W-property for .
Remark 11 We here adopt the de…nitions of a Bewley and of a justi…able preference set forth and discussed at more length in Cerreia-Vioglio and Ok (2018, Sect. 6). Let Rn be endowed with the natural topology and denote a nonempty, closed and convex subset of the simplex n 1 Rn. ABewley (a justi…able) preference with a prior set is a relationB onRn de…ned by
B(x) =fy2Rn : y xfor all (for some) 2 g.
Putting H = fz 2 Rn : z 0g for all 2 and C = T
2 H (and C = S
2 H ), it is readily checked that a relation B on Rn is a Bewley (a justi…able) preference with a prior set if and only if
B(x) =x+C for allx2Rn.
Therefore, a relation B on Rn is a Bewley (a justi…able) preference with prior set only if it coincides with the relation induced by the intersection (the union) of a family of half-spaces of Rn that are closed for the natural topology ofRn. Said this, part 1 (part 2) of Corollary 7 implies that a Bewley (a justi…able) preference is a relation with the strong (the weak) W-property.13
1 3Bewley (justi…able) preferences can be de…ned in the more general setting speci…ed in f.n.
27 of Cerreia-Vioglio and Ok (2018). Part 1 (part 2) of Corollary 7 implies that a Bewley (a justi…able) preference is a relation with the strong (the weak) W-property also in that setting:
the argument that leads to such a conclusion is essentially the same as that just exposed.